MinT - Best Known (81, 81+53, s)-Nets in Base 2

Best Known (81, 81+53, s)-Nets in Base 2

(81, 81+53, 60)-Net over F₂ — Constructive and digital

Digital (81, 134, 60)-net over F₂, using

2 times m-reduction [i] based on digital (81, 136, 60)-net over F₂, using
- trace code for nets [i] based on digital (13, 68, 30)-net over F₄, using
  - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
      - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(81, 81+53, 68)-Net over F₂ — Digital

Digital (81, 134, 68)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(81, 81+53, 328)-Net in Base 2 — Upper bound on s

There is no (81, 134, 329)-net in base 2, because

1 times m-reduction [i] would yield (81, 133, 329)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 10894 958769 005347 043393 829854 654613 694000 > 2¹³³ [i]